I use the following mnemonic devices to help students remember the steps required in carrying
out a hypothesis test or confidence interval: PHANTOMS and PANIC.
P
H
A
N
T
O
M
S
arameter
ypotheses
ssumptions
ame the test
est statistic
btain p-value
ake decision
tate conclusion in context
P
A
N
I
C
arameter
ssumptions
ame the interval
nterval
onclusion in context
Example:
A study of iron deficiency in infants compared samples of infants whose mothers chose different
ways of feeding them. One group contained breast-fed infants. The children in another group
were fed a standard baby formula without any iron supplements. Here are summary results on
blood hemoglobin levels at 12 months of age.
Group
Breast-Fed
Formula
n
23
19
x
13.3
12.4
s
1.7
1.8
Is there significant evidence that the mean hemoglobin level is different amoung breast-fed
babies? Give a 95% confidence interval for the mean difference in hemoglobin level between
the two populations of infants.
HYPOTHESIS TEST:
P
H
A
breast : the mean blood hemoglobin level at 12 months of age for breast-fed babies
 formula : the mean blood hemoglobin level at 12 months of age for formula-fed babies
H 0 : breast   formula
H 0 : breast   formula
It is reasonable to assume that the samples were chosen randomly and independently. It is also
reasonable to assume that at least 230 breast-fed babies and 190 formula-fed babies are present
in the population. Although we cannot check for outliers or strong skewness, the combined
sample size of 42 is sufficiently large.
N
Therefore, a two-sample t-test may be used.
T
t
O
df  37.5976
M
Do not reject H0.
S
There is not strong enough evidence to conclude that the mean blood hemoglobin level at 12
months of age for the breast-fed group is significantly different than that of the formula-fed
group. If there truly is no difference between breast-fed and formula-fed babies, we would
expect a result at least this extreme in about 10 out of every 100 samples due to chance.
x
breast
 x formula    breast   formula 
s 2breast s 2 formula

nbreast
n formula

13.3  12.4    0   1.6537
1.72 1.82

23
19
p  2P  t  1.6537   2  0.0533  0.1065
tcdf 1.6537,1E99,37.5976   0.0533
CONFIDENCE INTERVAL:
P
A
N
 : the mean difference in blood hemoglobin level at 12 months of age between breast-fed
babies and the formula-fed babies
It is reasonable to assume that the samples were chosen randomly and independently. It is also
reasonable to assume that at least 230 breast-fed babies and 190 formula-fed babies are present
in the population. Although we cannot check for outliers or strong skewness, the combined
sample size of 42 is sufficiently large.
Therefore, a two-sample t-interval may be used.
x
breast
s 2breast s 2 formula
1.7 2 1.82
 x formula   t 

 13.3  12.4   2.021

 0.9  1.0999
nbreast
n formula
23
19
I
(-0.2021, 2.0021)
C
We are 95% confident that the true mean level of hemoglobin at 12 months of age in breast-fed
babies is between about 0.2 units lower and 2.0 units higher than that of formula-fed babies,
since 95% of all samples of this size would produce a mean difference within 1.10 units of the
true mean difference.